This RMD will look in some detail at the items we’ve selected to look at in detail: meats, autos, staples.
The following code (not shown in markdown) will import and format the BEA data from csv tables. All of this is duplicated from Wavelet Analysis of Variance v2.
The following function (not shown in markdown) allows querying the main tables (exp, qua, and pri) based on item levels, where goods and services are level 1; durable goods, nondurable goods, and HH cons exp on services are level 2; and so on. Often, we’ll want the lowest (i.e. most granular level), which can be retrieved with the lowestLevel = T parameter. This is the same as in Process PCE Data.Rmd
This will build the itemClass table that’s the same in Wavelet Analysis of Variance v2.RMD.
## Item Pan.Price.Pct.Chg
## 35 Other recreational vehicles 15.32195
## 32 Bicycles and accessories 15.32225
## 34 Pleasure aircraft 15.32233
## 33 Pleasure boats 15.32300
## 154 Pension funds 15.45173
## 9 Tires 15.47034
## 54 Eggs 18.23457
## 11 Furniture 19.04585
## 48 Pork 19.83494
## 15 Major household appliances 23.73958
## 66 Food produced and consumed on farms (6) 24.02975
## 47 Beef and veal 24.47186
## 107 Natural gas (28) 30.58894
## 76 Other fuels 31.87270
## 6 Employee reimbursement 40.17855
## 118 Motor vehicle rental 40.17857
## 73 Gasoline and other motor fuel 43.68278
## 4 Net transactions in used autos 53.90803
## 7 Net transactions in used trucks 53.90829
## 75 Fuel oil 55.29619
## 8 Used truck margin 91.59348
## 5 Used auto margin 91.59380
## NA <NA> NA
## NA.1 <NA> NA
## NA.2 <NA> NA
## NA.3 <NA> NA
## NA.4 <NA> NA
## NA.5 <NA> NA
## NA.6 <NA> NA
## NA.7 <NA> NA
## NA.8 <NA> NA
## NA.9 <NA> NA
## NA.10 <NA> NA
## NA.11 <NA> NA
## NA.12 <NA> NA
## NA.13 <NA> NA
## NA.14 <NA> NA
## NA.15 <NA> NA
Whether using discrete or continuous wavelet transforms, wavelet analysis is, in essence, a way to characterize the “spectral characteristics of a time-series as a function of time, revealing how the different periodic components of a particular time-series evolve over time,” (Aguiar-Conraria and Soares 2011, 478). Whereas Fourier transforms show the frequency distribution of a whole series, wavelets can locate power at various frequencies for particular times in the series. This is done by projecting a wavelet at different scales, s, (widths) and translations, \(\tau\) (time locations) onto the series, \(x_t\). Hence, the continuous wavelet transform with respect to a chosen wavelet function \(\psi\) can be expressed as:
\[W_x(\tau, s) = \int x_t \left[\frac{1}{\sqrt{|s|}}\bar\psi \left(\frac{t-\tau}{s} \right) \right]\]
with the bar denoting complex conjugation (Auigar-Conraria and Soares 2011, 479). As we will ultimately be looking at synchronism between price and quantity for our Personal Consumption Expenditure items, an analytic wavelet is appropriate. As Aquiar-Conraria and Soares (2011, 479) note, “Analytic wavelets are ideal for the analysis of oscillatory signals, since the continuous analytic wavelet transform provides an estimate of the instantaneous amplitude and instantaneous phase of the signal in the vicinity of each time/scale location (\(\tau\), s).” Following those same authors, we utilize the Morlet wavelet, given by
\[\psi_{\omega_0}(t) = \pi^{-1/4}e^{i\omega_0t}e^{-\frac{t^2}{2}}\]
See Aquiar-Conraria and Soares (2011, 479) for the advantages of using this wavelet.
Once the wavelet is chosen, it is possible to calculate wavelet power spectra showing the distribution of variance in terms of both scale (frequency) and time. Furthermore, wavelet coherency between two time-series can be calculated (see Aguiar-Conraria and Soares 2011, 479-80 for more information). Wavelet coherency can be understood as an indication of correlation, in our case between price and quantity, in a particular period of time and for a particular frequency (e.g. price and quantity fluctuatios occuring over a periods of 2 months versus 2 years).
Here are the Continuous Wavelet Transform plots for those items.
[This takes 16 minutes to run at 10 simulations. I’ll run it at 500 simulations.]
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